Page 167 - Design of Reinforced Masonry Structures

DESIGN OF REINFORCED MASONRY BEAMS 4.31

Hence, steel has yielded and the assumption e ≥ 1.5e is valid.
s
y
φM = 17.66 k-ft
n
φ M 17 66
.
M = φ n = 09 = 19 62. k-ft > 1 3. M = 16 64. k-fft
cr
n
.
The nominal strength of the beam (19.62 k-ft) is greater than 1.3 times the crack-
ing moment of the beam (16.64 k-ft). Therefore, this beam satisfies the code
requirements for the nominal cracking moment strength of the beam.

4.7 DESIGN OF MASONRY BEAMS

4.7.1 Ductility in Reinforced Concrete Beams: Balanced, Underreinforced,
and Overreinforced Beams
Analysis of reinforced masonry beams presented in the preceding section is based on the
premise that crushing of masonry and yielding of steel reinforcement in the beam occur
simultaneously, the strain in masonry being e and the strain in the reinforcement as e .
mu
y
Such a state of strain is defined as a balanced condition, and the corresponding reinforce-
ment ratio is referred to as the balanced steel ratio, r . In a balanced beam, crushing of
b
masonry would occur without a warning, resulting in a catastrophic failure. Therefore, it is
highly desirable to provide an upper limit on the reinforcement ratio to ensure some degree
of ductility in beams, that is, to cause the steel reinforcement to yield before the onset of
crushing of masonry. Such beams are called underreinforced beams. Failures of such beams
are characterized by excessive deflection accompanied by strains in steel reinforcement
beyond the yield strain, e , thus exhibiting ductile behavior. This type of behavior provides
y
ample warning before the compression failure of masonry occurs. Accordingly, only under-
reinforced beams are permitted by the design codes. For practical purposes, the balanced
beam should be considered only as a hypothetical beam.

4.7.2 Determination of Balanced Steel Ratio, q b
Balanced condition is defined by a set of two strain values: strain in masonry at the time of
crushing, e , and the yield strain in steel, e . Crushing strain in masonry is set at 0.0025 for
y
mu
concrete masonry and 0.0035 for clay masonry [MSJC Code Section 3.3.2.(c)]. The value

of the balanced reinforcement ratio, r , can be derived in terms of the design strengths of
b
concrete and reinforcement. It is assumed that the neutral axis can be located by the linear
strain relationship as shown in Fig. 4.6. Based on this assumption, the relationship between
strains in masonry and reinforcement was derived earlier as given by Eq. (4.14):
−
ε s = dc
ε c (4.14 repeated)
mu
Equation (4.14) can be rearranged and expressed as
c ε
= mu (4.55)
d ε + ε
mu s